Question

Suppose Tom likes both ketchup (K) and mustard (M) on his hotdogs: U(K,M)=K2M. He spends $12 a month on these condiments. As usual in this class, assume items are infinitely divisible (i.e. Tom can buy any fraction of a ketchup or mustard bottle he chooses).

a. Initially, bottles of ketchup and mustard are equally priced at $2 per bottle. Find Tom’s optimal basket of condiments, and label these values KA and MA.

b. After an unusual drought that decimated tomato crops, the price of ketchup increases to $4 a bottle. Find Tom’s optimal basket now – KC, MC.

c. Now solve for the compensated demand response: fixing utility at the original level, what bundle (??, ??) will Tom consume if his budget could change, but he still wanted to spend as little as possible? Remember you can find this bundle by setting the tangency condition and the utility condition: ?(??, ?? ) = ?(??, ?? )

(Feel free to skip a and b as I only need the solution to c)

Answer 1

Consider the given utility function the ordinary demand functions are given by.

=> K = 2Y/3Pk, where “Y=income” and M = Y/3Pm. So, given the values the optimum consumption bundles are given by, “K*=2*12/3*2 = 4” and “M*=12/3*2 = 2”. So, the corresponding maximum utility is given by.

=> U = K^2*M= 4^2*2 = 32, => the maximum utility is given by “32”.

Now, the price of “K” changes to “4”, => “Pk=4” and “Pm=2”.

Now, the utility function is given by, “U=K^2*M”, => “MUk = 2KM” and “Mum = K^2”.

=> MRS = MUk/Mum = 2KM/K^2 = 2M/K = MRS. Now, at the equilibrium the relative price must be equal to the MRS, => MRS=Pk/Pm”.

=> 2M/K = Pk/Pm, => M = (Pk/2Pm)*K. The utility function is given by.

=> U=K^2M, => U=(Pk/2Pm)*K^3, => U*(2Pm/Pk) = K^3, => U^1/3*(2Pm/Pk)^1/3 = K, be the compensated demand function for “K”.

=> M = (Pk/2Pm)*K= (Pk/2Pm)* U^1/3*(2Pm/Pk)^1/3 = (Pk/2Pm)^2/3* U^1/3 = M, be the compensated demand function for “M”.

So, the optimum consumption to original level of utility is given by as follows.

=> M = (Pk/2Pm)^2/3*U^1/3= (4/2*2)^2/3*(32)^1/3= (32)^1/3 = 10.67 = M and.

=> K = (2Pm/Pk)^1/3*U^1/3 = (2*2/4)^1/3*32^1/3 = 10.67 = K, the optimum consumption to get the initial level of utility .